Intersection of two lines
How to find the relationship between two lines in the space.
To find the relationship between the line
|
A1x+B1y+C1z+D1=0 |
A2x+B2y+C2z+D2=0 |
and the line
|
A3x+B3y+C3z+D3=0 |
A4x+B4y+C4z+D4=0 |
Form a system with the equations and calculate the ranks.
r = rank of the coefficient matrix
r'= rank of the augmented matrix
The relationship between the line and the plane can be described as follow:
Case 1. Skew lines
r=3 and r'=4
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Case 2. Intersecting lines
r=3 and r'=3
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| |
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Case 3. Parallel lines
r=2 and r'=3
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Case 4. Coincident lines
r=2 and r'=2
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State the relationship between the following lines:
and 
Solution:
Form the system of equations and calculate the ranks.
) |
 |
| 1 |
-2 |
0 |
1 |
| 0 |
1 |
-1 |
-1 |
| 1 |
1 |
0 |
0 |
| 1 |
0 |
1 |
0 |
|
|
 |
r'=4 |
They are skew lines.
State the relationship between the following lines:
and 
Solution:
Form the system of equations and calculate the ranks.
) |
|
| 1 |
1 |
1 |
3 |
| 2 |
-1 |
1 |
2 |
| 1 |
-2 |
0 |
-1 |
| 3 |
0 |
-2 |
1 |
|
|
=0 |
r=3 |
They are Intersecting Lines.
